Example 29.43.2. Let $S$ be a scheme. Let $\mathcal{A}$ be a quasi-coherent graded $\mathcal{O}_ S$-algebra generated by $\mathcal{A}_1$ over $\mathcal{A}_0$. Assume furthermore that $\mathcal{A}_1$ is of finite type over $\mathcal{O}_ S$. Set $X = \underline{\text{Proj}}_ S(\mathcal{A})$. In this case $X \to S$ is projective. Namely, the morphism associated to the graded $\mathcal{O}_ S$-algebra map

$\text{Sym}_{\mathcal{O}_ X}^*(\mathcal{A}_1) \longrightarrow \mathcal{A}$

is a closed immersion, see Constructions, Lemma 27.18.5.

Comment #8109 by Laurent Moret-Bailly on

This is repeated later as Lemma 31.30.5. But the condition on $\mathcal{A}$ can be relaxed, at least if $S$ is quasicompact: by 10.56.2 and 27.11.8 it is enough to assume that $\mathcal{A}$ is finitely generated over $\mathcal{A}_0$ and $\mathcal{A}_0$ is finite over $\mathcal{O}_S$. This is useful in practice, but I could not find this statement in SP (or in EGA!).

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